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Galerkin approximations of nonlinear optimal control problems in Hilbert spaces
Mickaël D. Chekroun, Axel Kröner, Honghu Liu
To cite this version:
Mickaël D. Chekroun, Axel Kröner, Honghu Liu. Galerkin approximations of nonlinear optimal control problems in Hilbert spaces . Electronic Journal of Differential Equations, Texas State University, Department of Mathematics, 2017, 2017 (189), pp.1-40. �hal-01501178�
HILBERT SPACES
MICKAËL D. CHEKROUN, AXEL KRÖNER, AND HONGHU LIU
ABSTRACT. Nonlinear optimal control problems in Hilbert spaces are considered for which we derive approximation theorems for Galerkin approximations. Approximation theorems are available in the literature. The originality of our approach relies on the identification of a set of natural assumptions that allows us to deal with a broad class of nonlinear evolution equations and cost functionals for which we derive convergence of the value functions associated with the optimal control problem of the Galerkin approximations. This convergence result holds for a broad class of nonlinear control strategies as well.
In particular, we show that the framework applies to the optimal control of semilinear heat equations posed on a general compact manifold without boundary. The framework is then shown to apply to geoengineering and mitigation of greenhouse gas emissions formulated for the first time in terms of optimal control of energy balance climate models posed on the sphereS2.
CONTENTS
1. Introduction 1
2. Galerkin approximations for optimal control problems: Convergence results 4
2.1. Preliminaries 4
2.2. Convergence of Galerkin approximations: Trajectory-wise result 5
2.3. Local-in-uapproximation result 8
2.4. Convergence of Galerkin approximations: Uniform-in-uresult 11 2.5. Galerkin approximations of optimal control and value functions: Convergence results 15 2.6. Galerkin approximations of optimal control and value functions: Error estimates 18
2.7. Examples that satisfy Assumption(A7) 21
3. Application to optimal control of energy balance climate models 22
3.1. Preliminary from differential geometry 22
3.2. Galerkin approximations of controlled semilinear heat equations onSn 24
3.3. Energy balance models (EBMs) 26
3.4. Optimal control of climate? 28
3.5. Optimal control of EBMs: Convergence results of value functions 29
4. Concluding remarks 31
Acknowledgments 32
Appendix A. Existence of optimal controls 32
References 33
1. INTRODUCTION
Optimal control problems of infinite dimensional systems play an important role in a broad range of applications in engineering and various scientific disciplines [Lio71,Fat99,Fur00,BDPDM07,HPUU09,
1
Trö10]. Various methods for solving numerically the related optimization problems are available; see e.g. [HPUU09,MTZ08]. The case of linear evolution equations has benefited from a long tradition, and an abundant literature exists about finite element techniques or Galerkin methods for the design of approximate optimal controls; see e.g. [MBJ73,Gib79,Las80,Mal82,Kno82,BK84,LT87,AM89,LT00].
The case of Galerkin approximations of optimal control problems for nonlinear evolutions seems to have been much less addressed. Semidiscrete Ritz-Galerkin approximations of nonlinear parabolic boundary control problems have been considered for which convergence of the approximate controls have been obtained; see [Trö93,Trö94]. We refer also to [MV07,NV12] for error estimates concerned with space-time finite element approximations of the state and control to optimal control problems governed by semilinear parabolic equations, and to [DH04] for finite element approximations of op- timal control problems associated with the Navier-Stokes equations.
In this article, we study Galerkin approximations for (possibly non-quadratic) optimal control problems over a finite horizon[0,T], of nonlinear evolution equations in Hilbert space. Our frame- work covers not only a broad class of semilinear parabolic equations but also includes systems of nonlinear delay differential equations (DDEs) [CGLW16] and allows in each case for a broad class of nonlinear control strategies. The main contribution of this article is to identify for such equations a set of easily checkable conditions in practice, from which we prove the pointwise convergence of the value functions associated with the optimal control problem of the Galerkin approximations, and for a broad class of cost functionals; see Theorem2.2, our main result. This convergence at the level of value functions results essentially from a double uniform convergence—with respect to time and the set of admissible controllers—of the controlled Galerkin states; see Theorem2.1and Corollary2.1.
The treatment adopted here is based on the classical Trotter-Kato approximation approach from the C0-semigroup theory [Gol85,Paz83], which can be viewed as the functional analysis operator version of the Lax equivalence principle.1Within this approach, we generalize, in particular, the convergence results about value functions obtained in the earlier work [Fer97] concerned with the Galerkin ap- proximations to optimal control problems governed by linear evolution equations in Hilbert space.
Given a Hilbert state space H, denoting by ΠN the orthogonal projector associated with the N- dimensional Galerkin subspace, and byyN(·;ΠNx,u)the controlled Galerkin state (driven byu) and emanating fromΠNx, a key property to ensure convergence of the value functions for such optimal control problems is the following double uniform convergence
(1.1) lim
N→∞ sup
u∈Uad
sup
t∈[0,T]
kyN(t;ΠNx,u)−y(t;x,u)kH =0, ∀ x∈ H, whereUaddenotes a set of admissible controls; see [Fer97, Theorem 4.2].
When the evolution equation involves state- or control-dependent nonlinear terms, the conditions provided in [Fer97, Proposition 2.1] to ensure (1.1) needs to be amended. Whether the nonlinear terms involve the controls or the system’s state, our working assumption regarding the linear terms of the original equation and of its Galerkin approximations, is (as in [Fer97]) to satisfy respectively the
“stability” and “consistency” conditions required in the Trotter-Kato theorem; see Assumption(A1) and Assumption(A2)in Sect.2.1. In the case of a linear equation with nonlinear control terms, a sim- ple compactness assumption about the set of admissible controls (see Assumption(A5)in Sect.2.4) is sufficient to ensure (1.1); see Remark2.2.
In the case of an evolution equation depending nonlinearly on the system’s state and on the con- trols, a key assumption is introduced to ensure (1.1) that adds up to standard local Lipschitz condi- tions on the state-dependent nonlinear terms (Assumption(A3)) and the nonlinear control operator C:V → H, whereVdenotes an auxiliary Hilbert space in which the controls take values. Introduc- ing Π⊥N := IdH−ΠN, this assumption concerns a double uniform convergence about the residual
1i.e., if “consistency” and “stability” are satisfied, then “convergence” holds, and reciprocally.
energykΠ⊥Ny(t;x,w)kH(see Assumption(A7)in Sect.2.4), namely
(1.2) lim
N→∞ sup
u∈Uad
sup
t∈[0,T]
kΠ⊥Ny(t;x,u)kH=0.
With this assumption at hand, and the rest of our working assumptions, standarda prioribounds (Assumption (A6)) allow us to ensure (1.1) for a broad class of nonlinear evolution equations in Hilbert spaces. The pointwise convergence of the value functions associated with the optimal control problem of the corresponding Galerkin approximations is then easily derived for a broad class of cost functionals; see Theorem2.2.
The relevance of assumption (1.2) for applications is addressed through various angles. First, from the proof of Corollary2.1(and thus Theorem2.1) in which Theorem2.2relies. In that respect, a sort of pedagogical detour is made in Sect.2.3in which we show essentially that a weaker (than (1.1)) local- in-u approximation result (Lemma 2.3) follows from the rest of our working assumptions (except Assumptions(A6)and(A7)2) and from a local-in-uestimate about the residual energy (Lemma2.1);
the latter resulting itself from the continuity of the mappingu7→y(t;x,u). Condition (1.2) constitutes thus a natural strengthening of inherent properties to the approximation problem.
From a more applied perspective, sufficient conditions concerning the spectrum of the linear part—such as self-adjointness and compact resolvent—are pointed out in Sect. 2.7 to ensure (1.2);
see Lemma2.6and Remark2.6. Finally, Sect.2.6provides error estimates concerning the value func- tion and the optimal control that complete the picture and emphasize from another perspective the relevance of the residual energy in the analysis of the approximation problem; see Theorem2.3and Corollary2.2.
With this preamble in mind, we provide now the more formal organization of this article. First, we present in Sect. 2.1 the type of state equation and its corresponding Galerkin approximations that we will be working on. A trajectory-wise convergence result for each fixed control u is then derived in Sect.2.2. As mentioned earlier, it relies essentially on the theory ofC0-semigroups and the Trotter-Kato theorem [Paz83, Thm. 4.5, p.88]; see Lemma2.1. In a second step, we derive a “local- in-u” approximation result in Sect.2.3for controls that lie within a neighborhood of a given control u; see Lemma2.3. As discussed above, a key approximation property about the residual energy of solutions (see (2.33)) is then amended into an assumption (see Assumption(A7)) to ensure a uniform- in-uconvergence result; see Theorem2.1of Sect.2.4. As shown in Sect.2.5, this uniform convergence result helps us derive—in the spirit of dynamic programming (see Corollary2.1)—the convergence of the value functions associated with optimal control problems based on Galerkin approximations;
see Theorem2.2. For this purpose, some standard sufficient conditions for the existence of optimal controls are also recalled in AppendixA. Simple and useful error estimates about the value function and the optimal control are then provided in Sect. 2.6. In Sect. 2.7 we point out a broad class of evolution equations for which Assumption(A7)is satisfied.
As applications of the theoretical results derived in Sect. 2, we show in Sect.3 that our frame- work allows to provide rigorous Galerkin approximations to the optimal control of a broad class of semilinear heat problems, posed on a compact (smooth) manifold without boundary. As a concrete example, the framework is shown to apply to geoengineering and the mitigation of greenhouse gas (GHG) emissions formulated for the first time here in terms of optimal control of energy balance models (EBMs) arising in climate modeling; see [Bud69,Sel69,Ghi76,NCC81] for an introduction on EBMs, and [DH98,BCDT09] for a mathematical analysis. After recalling some fundamentals of differ- ential geometry in Sect.3.1to prepare the analysis, a general convergence result of Galerkin approx- imations to controlled semilinear heat problems posed on ann-dimensional sphereSnis formulated
2More precisely, by assuming a weaker version of Assumption(A6), namely Assumption(A4), and without assuming (A7).
in Sect.3.2; see Corollary3.1. The application to the optimal control of EBMs is then presented in Sect.3.3in the context of geoengineering and GHG emissions for which approximation of the value function and error estimates about the optimal control are obtained. Finally, Sect.4outlines several possible directions for future research the framework introduced in this article opens up.
2. GALERKIN APPROXIMATIONS FOR OPTIMAL CONTROL PROBLEMS: CONVERGENCE RESULTS
We present in this section, rigorous convergence results for semi-discretization of optimal control problems based on Galerkin approximations. In particular, we derive the pointwise convergence of the value functions associated with optimal control problems based on Galerkin approximations in Sect.2.5.
2.1. Preliminaries. We consider in this section finite-dimensional approximations of the following initial-value problem (IVP):
(2.1)
dy
dt = Ly+F(y) +C(u(t)), t∈ (0,T], y(0) =x,
wherexlies inH, andHdenotes a separable Hilbert space. The time-dependent forcingulives in a separable Hilbert spaceV(possibly different thanH); the (possibly nonlinear) mappingC: V → H is assumed to be such that C(0) = 0. Other assumptions regarding C will be made precise when needed.
We assume that the linear operator L : D(L) ⊂ H → H is the infinitesimal generator of a C0- semigroup of bounded linear operatorsT(t)on H. Recall that in this case the domainD(L)of Lis dense inHand thatLis a closed operator; see [Paz83, Cor. 2.5, p. 5].
Under the above assumptions on the operatorL, recall that there existsM ≥ 1 andω ≥0 [Paz83, Thm. 2.2, p. 4] such that
(2.2) kT(t)k ≤Meωt, t≥0,
wherek · kdenotes the operator norm subordinated tok · kH. For the moment, we take the set of admissible controls to be
(2.3) U := Lq(0,T;V),
withq≥ 1. In the later subsections, further assumptions on the admissible controls will be specified when needed.
Letu be inU given by (2.3), amild solutionto (2.1) over[0,T]is a functionyin C([0,T],H)such that
(2.4) y(t) =T(t)x+
Z t
0 T(t−s)F(y(s))ds+
Z t
0 T(t−s)C(u(s))ds, t∈ [0,T]. In what follows we will often denote byy(s;x,u)a mild solution to (2.1).
Let{HN :N∈Z∗+}be a sequence of finite-dimensional subspaces ofHassociated withorthogonal projectors
(2.5) ΠN :H → HN,
such that
(2.6) k(ΠN−Id)xk −→
N→∞0, ∀x∈ H, and
(2.7) HN ⊂ D(L), ∀N≥1.
The corresponding Galerkin approximation of (2.1) associated withHNis then given by:
(2.8)
dyN
dt =LNyN+ΠNF(yN) +ΠNC(u(t)), t∈[0,T], yN(0) =ΠNx, x ∈ H,
where
(2.9) LN :=ΠNLΠN :H → HN.
In particular, the domainD(LN)ofLN isH, because of (2.7).
Throughout this section, we assume the following set of assumptions:
(A0) The linear operator L : D(L)⊂ H → His the infinitesimal generator of a C0-semigroup of bounded linear operators T(t)onH.
(A1) For each positive integer N, the linear flow eLNt :HN → HNextends to a C0-semigroup TN(t)onH. Furthermore the following uniform bound is satisfied by the family{TN(t)}N≥1,t≥0
(2.10) kTN(t)k ≤ Meωt, N≥1, t≥0,
wherekTN(t)k:=sup{kTN(t)xkH, kxkH≤1,x∈ H}and the constants M andωare the same as given in(2.2).
(A2) The following convergence holds
(2.11) lim
N→∞kLNφ−LφkH=0, ∀φ∈ D(L). (A3) The nonlinearity F is locally Lipschitz in the sense given in(2.12)below.
Following the presentation commonly adopted for the Trotter-Kato approach, the assumptions (A0)-(A2) are concerned with the linear parts of the original system (2.1) and of its Galerkin ap- proximation (2.8). Assumption(A3)is concerned with the nonlinearity in (2.1). Other assumptions regarding the latter will be made in the sequel. Throughout this article, a mapping f : W1 → W2 between two Banach spaces,W1andW2, is said to be locally Lipschitz if for any ballBr ⊂ W1with radiusr >0 centered at the origin, there exists a constant Lip(f|Br)>0 such that
(2.12) kf(y1)− f(y2)kW2 ≤Lip(f|Br)ky1−y2kW1, ∀y1,y2 ∈Br.
2.2. Convergence of Galerkin approximations: Trajectory-wise result. As a preparation for the main result given in Sect. 2.5, we derive hereafter a trajectory-wise convergence result for the so- lutions to the Galerkin approximations (2.8); see Lemma2.1below.
With this purpose in mind, besides(A0)–(A3), we will also make use of the following assumption.
(A4) For each T > 0, x inH and each u inU withU defined in(2.3), the problem(2.1)admits a unique mild solution y(·;x,u)in C([0,T],H), and its Galerkin approximation(2.8)admits a unique solution yN(·;ΠNx,u)in C([0,T],HN)for each N inZ∗+. Moreover, there exists a constant C :=C(T,x,u) such that
(2.13) kyN(t;ΠNx,u)kH≤C, ∀t ∈[0,T], N∈ Z∗+.
Note that in applications,(A4)is typically satisfied viaa prioriestimates; see Remark2.1-(ii) below.
Lemma 2.1. LetU be the set of admissible controls given by(2.3), with q defined therein. Consider the IVP (2.1) and the associated Galerkin approximation (2.8). Assume that the nonlinear operator C : V → H satisfies, for1≤ p≤q, the following growth condition,
(2.14) kC(w)kH≤ γ1kwkVp +γ2, ∀w∈V, whereγ1 >0andγ2≥0.
Assume also that(A0)–(A4)hold. Then for each fixed T>0, x inHand u inU, the following convergence result is satisfied:
(2.15) lim
N→∞ sup
t∈[0,T]
kyN(t;ΠNx,u)−y(t;x,u)kH=0.
Proof. To simplify the notations, only thex-dependency that matters to the estimates derived here- after will be made explicit. Letube given inUandxinH, then by the variation-of-constants formula applied to Eq. (2.8) we have, for 0≤t≤ TandNinZ∗+, that
(2.16) yN(t;u) =eLNtΠNx+
Z t
0 eLN(t−s)ΠNF(yN(s;u))ds+
Z t
0 eLN(t−s)ΠNC(u(s))ds.
Then, it follows from (2.4) and (2.16) that the differencewN(t;u):=y(t;u)−yN(t;u)satisfies
(2.17)
wN(t;u) =T(t)x−eLNtΠNx+
Z t
0 T(t−s)F(y(s;u))ds−
Z t
0 eLN(t−s)ΠNF(yN(s;u))ds +
Z t
0 T(t−s)C(u(s))ds−
Z t
0 eLN(t−s)ΠNC(u(s))ds and hence, we have
(2.18)
wN(t;u) =T(t)x−eLNtΠNx+
Z t
0 T(t−s)−eLN(t−s)ΠN
F(y(s;u))ds +
Z t
0 eLN(t−s)ΠN F(y(s;u))−F(yN(s;u))ds+
Z t
0
T(t−s)−eLN(t−s)ΠN
C(u(s))ds.
Let us introduce for everysin[0,T],xinH, anduinU, rN(s;u):= ky(s;u)−yN(s;u)kH, (2.19a)
ζN(x):= sup
t∈[0,T]
kT(t)x−eLNtΠNxkH, (2.19b)
dN(s;u):= sup
t∈[s,T]
k T(t−s)−eLN(t−s)ΠN
F(y(s;u))kH, (2.19c)
and for almost everysin[0,T],
(2.20) deN(s;u):= sup
t∈[s,T]
k T(t−s)−eLN(t−s)ΠN
C(u(s))kH.
ForxinHanduinU, let us denote byBthe closed ball inHwith radiusCcentered at the origin whereCis the upper bound in estimate (2.13) for the Galerkin solutions (Assumption(A4)).
Since by assumption,t7→ y(t;x,u)lies inC([0,T],H)for anyxinHanduinU given by (2.3), one can assume without loss of generality thaty(t;x,u)stays inBfor alltin[0,T], by possibly redefining C.
We obtain then from (2.18) that (2.21)
rN(t;u)≤ ζN(x) +
Z t
0
dN(s;u)ds+
Z t
0
keLN(t−s)ΠN F(y(s;u))−F(yN(s;u))kHds+
Z t
0
deN(s;u)ds
≤ ζN(x) +
Z t
0
dN(s;u)ds+MLip(F|B)
Z t
0
eω(t−s)rN(s;u)ds+
Z t
0
deN(s;u)ds
≤ ζN(x) +
Z t
0
dN(s;u)ds+MLip(F|B)eωT Z t
0
rN(s;u)ds+
Z t
0
deN(s;u)ds,
where we have used that Fis locally Lipschitz (Assumption(A3)) and the uniform bound (2.10) of Assumption(A1).
It follows then from Gronwall’s inequality that (2.22) rN(t;u)≤ ζN(x) +
Z T
0 dN(s;u)ds+
Z T
0
deN(s;u)ds
exp MLip(F|B)eωTT
, ∀t∈[0,T]. We are thus left with the estimation ofζN(x),RT
0 dN(s;u)dsandRT
0 deN(s;u)dsasN→∞. Note that the assumptions(A1)and(A2)allow us to use a version of the Trotter-Kato theorem [Paz83, Thm. 4.5, p.88] which implies together with (2.6) that
(2.23) lim
N→∞eLNtΠNφ= T(t)φ, ∀φ∈ H, uniformly intlying in bounded intervals; see also [CGLW16, footnote 6].
It follows that
(2.24) lim
N→∞ζN(x) =0,
and thatdN(·;u)anddeN(·;u)converge point-wisely to zero on[0,T], i.e.,
(2.25) lim
N→∞dN(s;u) =0, ∀s ∈[0,T], and
(2.26) lim
N→∞deN(s;u) =0, ∀s ∈[0,T].
On the other hand, by using (2.2) and (2.10) and from the local Lipschitz assumption onF, we get
(2.27)
k T(t−s)−eLN(t−s)ΠN
F(y(s;u))kH
≤2Meω(t−s)kF(y(s;u))kH
≤2Meω(t−s)
kF(y(s;u))−F(0)kH+kF(0)kH
≤2Meω(t−s) Lip(F|B)ky(s;u)kH+kF(0)kH, 0≤s ≤t≤ T, which implies
(2.28) dN(s,u)≤2MeωT Lip(F|B)ky(s;u)kH+kF(0)kH, ∀s∈ [0,T].
Sincey(·;u)lies inC([0,T];H), the mappings 7→ ky(s,u)kHis in particular integrable on[0,T], and the Lebesgue dominated convergence theorem allows us to conclude from (2.25) and (2.28) that
(2.29) lim
N→∞ Z T
0 dN(s;u)ds=0.
Let us estimateRT
0 deN(s;u)dsasN→∞. By using the growth condition (2.14), we get (2.30)
k T(t−s)−eLN(t−s)ΠN
C(u(s))kH≤2MeωT
γ1ku(s)kVp +γ2
, for a.e. s∈[0,T]and allt ∈[s,T]. Since u lies in Lq([0,T];V)with q ≥ p, thenu lies in Lp([0,T];V) and the right hand side (RHS) of (2.30) is integrable on[0,T]. The Lebesgue dominated convergence theorem allows us then to conclude from (2.26) and (2.30) that
(2.31) lim
N→∞ Z T
0
deN(s;u)ds=0.
The desired convergence result (2.15) follows now from (2.22) by using (2.24), (2.29) and (2.31).
2.3. Local-in-u approximation result. In this section, we consider for q ≥ 1, Uad, to be the subset of Lq(0,T;V), constituted by measurable functions that take values in U, a bounded subset of the Hilbert spaceV. In other words,
(2.32) Uad:={f ∈ Lq(0,T;V) : f(s)∈Ufor a.e. s∈[0,T]}, q≥1.
The setUadwill be endowed with the induced topology from that ofLq(0,T;V).
We present hereafter a natural property that is derived from our working assumptions, namely that given an finite-dimensional approximationHN ofH, the residual energy of solutions to the IVP (2.1), i.e.
k(IdH−ΠN)y(t;x,w)kH,
can be made small as N → ∞and uniformly inw, provided thatw lies within a sufficiently small open set ofUadgiven in (2.32).
This is the purpose of Lemma 2.2 which boils down to proving the continuity of the mapping u 7→ y(t;x,u); see (2.45) below. As a consequence a local-in-u approximation result is naturally inferred; see Lemma2.3. However, as pointed out and amended in Sect.2.4below, this is insufficient to guarantee convergence results for the value functions associated with Galerkin approximations of optimal control problems subordinated to (2.1).
The merit of Lemma2.2below is nevertheless not only to identify a symptom, but also to help us propose a cure. Indeed, by requiring the residual energy of the solution to the IVP (2.1) to vanish uniformly (in u) as N → ∞, we are able to conclude about the desired convergence results for the value functions. The latter uniform property (i.e. the “cure”) is shown below to hold for a broad class of IVPs; see Sect.2.7. For the moment, let us present the “symptom,” i.e. the local-in-uapproximation results. For that purpose, we start with a local-in-uestimate about the residual energy,
k(IdH−ΠN)y(t;x,w)kH, wherewlives in some neighborhood ofu.
Lemma 2.2. Assume that(A0)–(A4) hold. Assume furthermore thatC : V → His locally Lipschitz and that the admissible controls lie inUadgiven by(2.32)and endowed with the induced Lq(0,T;V)-topology, for q>1.
Then, for any(x,u)inH × Uad and anye > 0there exists a neighborhoodOu ⊂ Uadof u and N0 ≥ 0 such that the mild solution y(t;x,u)to(2.1)satisfies
(2.33) sup
w∈Ou
sup
t∈[0,T]
k(IdH−ΠN)y(t;x,w)kH≤ e, ∀N≥ N0.
Proof. As we will see, we mainly need to show that the solutiony(t;x,u)to the IVP (2.1) depends continuously on the control u in Uad, endowed with the Lq(0,T;V)-topology. The above estimate (2.33) follows then directly from this continuous dependence as explained at the end of the proof.
Given u in Uad andr > 0, we denote by BU
ad(u,r) the closed ball centered at u with radius r, for the induced Lq(0,T;V)-topology onUad. For anywinBU
ad(u,r),x inH andt in[0,T], consider φ(t):=y(t;x,u)−y(t;x,w)and note thatφ(0) =0. It follows from (2.4) that
(2.34) φ(t) =
Z t
0 T(t−s) F(y(s;x,u))−F(y(s;x,w))ds+
Z t
0 T(t−s) C(u(s))−C(w(s))ds.
We have then (2.35)
kφ(t)kH≤
Z t
0
Meω(t−s)kF(y(s;x,u))−F(y(s;x,w))kHds+
Z t
0
Meω(t−s)kC(u(s))−C(w(s))kHds.
LetC>0 be chosen such thatky(t;x,u)kH ≤Cfor allt∈[0,T], and letwbe inBUad(u,r), we define then
(2.36) t∗ :=max{t∈ [0,T] : ky(t;x,w)kH<2C}.
First let us note that t∗ > 0. Indeed recalling thatky(t;x,u)kH ≤ C by assumption, we have in particular thatkxkH≤C. Now due to the continuity for any(x,w)inH × Uadof the mapping
[0,T]−→ H t7−→y(t;x,w),
(sincey(t;x,w)is a mild solution), we infer, sincekxkH ≤C, for eachwinBU
ad(u,r)the existence of t0(w)>0 such that
ky(t;x,w)kH<2C, ∀t ∈[0,t0(w)], and thereforet∗ >0.
Denote alsoBH ⊂ Hthe closed ball centered at the origin with radius 2C. LetBV be the smallest closed ball inVcontaining the bounded setU. By using the local Lipschitz property ofFandC, we obtain from (2.35) that
(2.37) kφ(t)kH≤
Z t
0 Meω(t−s)Lip(F|BH)kφ(s)kHds+
Z t
0 Meω(t−s)Lip(C|BV)ku(s)−w(s)kVds
≤ MLip(F|BH)
Z t
0 eω(t−s)kφ(s)kHds+eωt∗MLip(C|BV)
Z t
0
ku(s)−w(s)kVds, ∀t∈ [0,t∗]. By Hölder’s inequality, we have
(2.38)
Z t
0
ku(s)−w(s)kVds≤tp
−1
p ku−wkLp(0,T;V)≤tp
−1 p r, which leads to
(2.39) kφ(t)kH ≤ MLip(F|BH)
Z t
0
eω(t−s)kφ(s)kHds+ (t∗)p−p1reωt∗MLip(C|BV), ∀t∈ [0,t∗]. It follows then from Gronwall’s inequality that
(2.40) kφ(t)kH≤ (t∗)
p−1
p rMLip(C|BV)exp(2ωt∗+t∗MLip(F|BH))
≤ Tp−p1rMLip(C|BV)exp(2ωT+TMLip(F|BH)), ∀t ∈[0,t∗]. Now, letC∗ := Tp
−1
p MLip(C|BV)exp(2ωT+TMLip(F|BH))and r1:= C
2C∗.
We claim thatt∗ =Tifr ≤r1. Otherwise, ift∗ <T, applying (2.40) att =t∗, we get
(2.41) kφ(t∗)kH ≤ C
2, which leads then to
(2.42) ky(t∗;x,w)kH ≤ ky(t∗;x,u)kH+kφ(t∗)kH ≤ 3 2C.
This last inequality contradicts with the definition oft∗ given by (2.36).
We obtain thus for eachr ∈(0,r1]that
(2.43) kφ(t)kH≤rC∗, ∀t ∈[0,T].
Now, it follows from (2.43) that for any fixede>0 there existsre>0 sufficiently small such that
(2.44) kφ(t)kH ≤ 1
2e, ∀t ∈[0,T]. Recalling the definition ofφ, we have thus proved that
(2.45) sup
w∈BUad(u,re)
sup
t∈[0,T]
ky(t;x,u)−y(t;x,w)kH≤ 1 2e.
We turn now to the last arguments needed to prove (2.33). It consists first to note that the conver- gence property (2.6) and the continuity oft 7→ y(t;x,u)imply, for the givene>0, the existence of a positive integerN0for which
(2.46) sup
t∈[0,T]
k(IdH−ΠN)y(t;x,u)kH≤ 1
2e, ∀N≥ N0. Now by definingΠ⊥N :=IdH−ΠN, and noting that
(2.47) kΠ⊥Ny(t;x,w)kH≤ kΠ⊥N(y(t;x,w)−y(t;x,u))kH+kΠ⊥Ny(t;x,u)kH
≤ ky(t;x,w)−y(t;x,u)kH+kΠ⊥Ny(t;x,u)kH,
we conclude—from (2.45) and (2.46)—to the desired estimate (2.33) withOutaken to be ˚BUad(u,re), the open ball inUadof radiusre.
We conclude this section with a local-in-uapproximation result.
Lemma 2.3. Assume the assumptions of Lemma2.2 hold. Then, for any(x,u)inH × Uad and anye > 0 there exists a neighborhoodOu⊂ Uadof u and N0≥1such that the mild solution y(t;x,u)to(2.1)satisfies
(2.48) sup
w∈Ou
sup
t∈[0,T]
kyN(t;ΠNx,w)−y(t;x,w)kH≤e, ∀N≥ N0, where yN denotes the solution to the Galerkin approximation(2.8).
Proof. First, let us remark that even if hereCdoes not satisfy the growth condition (2.14), one can still derive the trajectory-wise convergence result (2.15) by exploiting the fact thatu lies inUadandCis locally Lipschitz. The only change in the proof consists indeed of replacing the estimate (2.30) by the following:
(2.49)
k T(t−s)−eLN(t−s)ΠN
C(u(s))kH≤2MeωTLip(C|BV)ku(s)kV, for a.e.s∈ [0,T]and allt∈[s,T], where BV denotes the smallest closed ball in V containing the bounded set U. Since u lies in Lq([0,T];V), then u lies in L1([0,T];V) and the RHS of (2.49) is integrable on [0,T]. We can then still use the Lebesgue dominated convergence theorem to obtain (2.31) as before.
We explain now how to derive (2.48) from (2.15). Recall that from (2.45) derived in the proof of Lemma 2.2, the mild solutions to the IVP (2.1) depend continuously on the control u in Uad, en- dowed with theLq([0,T];V)-topology. An estimate similar to (2.45) can be derived for the solutions to Galerkin approximation (2.8), ensuring thus also their continuous dependence onu.
As a consequence, for any givene>0 anduinUad, there exists a neighborhoodOu⊂ Uadcontain- ingufor which
(2.50)
sup
w∈Ou
sup
t∈[0,T]
ky(t;x,u)−y(t;x,w)kH ≤ 1 3e, sup
w∈Ou
sup
t∈[0,T]
kyN(t;ΠNx,u)−yN(t;ΠNx,w)kH≤ 1
3e, ∀N∈Z∗+.
On the other hand, the trajectory-wise convergence (2.15) ensures the existence of a positive integer N0, such that
(2.51) sup
t∈[0,T]
kyN(t;ΠNx,u)−y(t;x,u)kH≤ 1
3e, ∀N ≥N0. Since
(2.52)
kyN(t;ΠNx,w)−y(t;x,w)kH≤ kyN(t;ΠNx,w)−yN(t;ΠNx,u)kH +kyN(t;ΠNx,u)−y(t;x,u)kH +ky(t;x,u)−y(t;x,w)kH, the desired estimate (2.48) follows from (2.50) and (2.51).
2.4. Convergence of Galerkin approximations: Uniform-in-u result. In the previous section, the local-in-u approximation result has been derived under a boundedness assumption on U arising in the definition ofUad. Here compactness will substitute the boundedness to derive convergence results that are uniform inu. More precisely, we will assume for that purpose
(A5) The set of admissible controls Uad is given by (2.32) with U being a compact subset of the Hilbert space V.
We will make also use of the following assumptions.
(A6) LetUad be given by(2.32). For each T > 0, (x,u)inH × Uad, the problem(2.1) admits a unique mild solution y(·;x,u)in C([0,T],H), and for each N ≥ 0, its Galerkin approximation(2.8)admits a unique solution yN(·;ΠNx,u)in C([0,T],H). Moreover, there exists a constantC := C(T,x)such that
(2.53) ky(t;x,u)kH≤ C, ∀t ∈[0,T], u ∈ Uad,
kyN(t;ΠNx,u)kH≤ C, ∀t ∈[0,T], N ∈Z∗+, u∈ Uad.
(A7) LetUadbe given by(2.32). For each fixed T >0, and any mild solution y(·;x,u)to(2.1)with(x,u) inH × Uad, it holds that
(2.54) lim
N→∞ sup
u∈Uad
sup
t∈[0,T]
k(IdH−ΠN)y(t;x,u)kH =0.
Remark 2.1.
(i) Note that(A6)differs from(A4)by the inequalityky(t;x,u)kH ≤ C, and that the constant in(2.53) is independent of the control u.
(ii) Let u be in Uad given by(2.32). Then uniform bounds such as in(2.53) are guaranteed if e.g. an a priori estimate of the following type holds for the IVPs(2.1)and(2.8):
(2.55) sup
t∈[0,T]
ky(t;x,u)kH≤α(kxkH+kukLq(0,T;V)) +β, α>0, β≥0.
(iii) We refer to Sect.2.7below for a broad class of IVPs for which Assumption(A7)holds.
As a preparatory lemma to Theorem2.1, we first prove that givenxinH, (IdH−ΠN)F(y(t;x,u)) −→
N→∞0,
uniformly inulying in Uadandt in[0,T]. For that only assumptions(A3),(A6)and(A7)are used, andUinvolved in the definition (2.32) ofUad is assumed to be bounded (not necessarily compact).
We have
Lemma 2.4. Assume that(A3),(A6)and(A7)hold. Then,
(2.56) lim
N→∞ sup
u∈Uad
sup
t∈[0,T]
k(IdH−ΠN)F(y(t;x,u))kH =0.
Proof. For any givene>0, by(A7), there existsN0inZ∗+such that
(2.57) sup
u∈Uad
sup
t∈[0,T]
k(IdH−ΠN)y(t;x,u)kH≤e, ∀N≥ N0. Note also that for allxinH, anduinUadgiven by (2.32)
(2.58) k(IdH−ΠN)F(y(t;x,u))kH≤ k(IdH−ΠN)F(ΠN0y(t;x,u))kH
+k(IdH−ΠN) F(y(t;x,u))−F(ΠN0y(t;x,u))kH. For the second term on the RHS, we have
(2.59)
k(IdH−ΠN) F(y(t;x,u))−F(ΠN0y(t;x,u))kH
≤ kF(y(t;x,u))−F(ΠN0y(t;x,u))kH
≤Lip(F|B)k(IdH−ΠN0)y(t;x,u)kH, ∀t ∈[0,T], u∈ Uad, whereBdenotes the ball inHcentered at the origin with radiusCgiven by (2.53).
It follows then from (2.57) that
(2.60) sup
u∈Uad
sup
t∈[0,T]
k(IdH−ΠN) F(y(t;x,u))−F(ΠN0y(t;x,u))kH ≤Lip(F|B)e.
Due to(A6), for eachxinH,ΠN0y(s;x,u)lies in a uniformly (insandu) bounded subset ofHN0. The finite dimensionality ofHN0 (as a Galerkin subspace) ensures the compactness of the set
Ex :={ΠN0y(s;x,u), u∈ Uad, s∈[0,T]}H.
For eachzinExande>0, due to (2.6) there exists an integer N1(z)such that k(IdH−ΠN)F(z)k ≤ e
2, ∀N≥ N1(z). SinceFis continuous, there exists a neighborhoodNz ofzinHN0 such that (2.61) k(IdH−ΠN)F(w)k ≤e, ∀N≥ N1(z), w∈ Nz.
From the compactness ofExwe can extract a finite cover ofEx by such neighborhoodsNz for which (2.61) holds, and thus one can ensure the existence of an integerN1for which
(2.62) sup
z∈Ex
k(IdH−ΠN)F(z)kH≤e, ∀N≥ N1.
This last inequality ensures for eachxinH, the existence an integerN1for which
(2.63) sup
u∈Uad
sup
t∈[0,T]
k(IdH−ΠN)F(ΠN0y(t;x,u))kH ≤e, ∀N≥ N1. Then, (2.56) follows from (2.58), (2.60) and (2.63).
We are now in position to formulate a uniform (inu) version of Lemma2.1 in which the growth condition (2.14) is no longer required.
Theorem 2.1. Assume that(A0)–(A3)and(A5)–(A7)hold. Assume also thatC : V → H is continuous.
Then, for any(x,u)inH × Uad, the mild solution y(t;x,u)to(2.1)satisfies the following uniform convergence result
(2.64) lim
N→∞ sup
u∈Uad
sup
t∈[0,T]
kyN(t;ΠNx,u)−y(t;x,u)kH=0, where yN denotes the solution to the Galerkin approximation(2.8).
Proof. Compared to Lemma2.1, we have replaced (A4) by the stronger assumption (A6) and the set of admissible controlsUadis as given in(A5). By following the proof of Lemma2.1, in order to obtain the uniform convergence result (2.64), it suffices to show that the two termsRT
0 dN(s;u)dsand RT
0 deN(s;u)dsinvolved in the RHS of (2.22) converge to zero asN →∞uniformly with respect tou inUad.
Recall from (2.19c) thatdN(s;u) =supt∈[s,T]k T(t−s)−eLN(t−s)ΠN
F(y(s;x,u))kH, which is de- fined for everysin[0,T]anduinUad. Thanks to Lemma2.4, for any fixede > 0, there existsN0 in Z∗+, such that
(2.65) k(IdH−ΠN0)F(y(s;x,u))kH<e, s ∈[0,T], u ∈ Uad. Now, forN0chosen above, we have
(2.66)
k T(t−s)−eLN(t−s)ΠN
F(y(s;x,u))kH≤ k T(t−s)−eLN(t−s)ΠN
ΠN0F(y(s;x,u))kH +k T(t−s)−eLN(t−s)ΠN
(IdH−ΠN0)F(y(s;x,u))kH. By using (2.2) and Assumption(A1), we obtain:
(2.67)
sup
t∈[s,T]
k T(t−s)−eLN(t−s)ΠN
(IdH−ΠN0)F(y(s;x,u))kH
≤2MeωTk(IdH−ΠN0)F(y(s;x,u))kH
≤2MeωTe, ∀s ∈[0,T], u∈ Uad, where the last inequality follows from (2.65).
Due to(A6), for eachx inH,y(s;x,u)lies in a uniformly (ins andu) bounded subset ofH. This together with the locally Lipschitz property ofFimplies thatΠN0F(y(s;x,u))lies in a bounded subset of HN0 for all u in Uad and for all s in [0,T]. The finite dimensionality of HN0 ensures then the compactness of the set
Ex0 = {ΠN0F(y(s;x,u)), u∈ Uad, s∈[0,T]}H.
Now, for eachzin Ex0 ande > 0, the convergence property (2.23) valid uniformly over bounded time-intervals allows us to ensure the existence of an integerN1(z), for which
sup
t∈[0,T]
k T(t)−eLNtΠN
zkH ≤ e
2, ∀ N ≥N1(z).
Then, by using (2.2) and Assumption(A1), there exists a neighborhoodNzofzinHN0 such that
(2.68) sup
t∈[0,T]
k T(t)−eLNtΠN
wkH≤ e, ∀ N ≥N1(z), w∈ Nz.
From the compactness ofEx0 we can extract a finite cover ofEx0 by such neighborhoods in which (2.68) holds, and thus one can ensure the existence of an integerN1for which
(2.69) sup
t∈[0,T]
k T(t)−eLN(t)ΠN
wkH ≤e, ∀N≥ N1, w∈ Ex0.
Now, for each fixedsin[0,T]anduinUad, by takingw=ΠN0F(y(s;x,u)), we get from (2.69) that sup
t∈[0,T]
k T(t)−eLN(t)ΠN
ΠN0F(y(s;x,u))kH≤ e, ∀N≥ N1. It follows then,
(2.70) sup
t∈[s,T]
k T(t−s)−eLN(t−s)ΠN
ΠN0F(y(s;x,u))kH≤ e, ∀N≥ N1, s ∈[0,T], u∈ Uad. By using (2.67) and (2.70) in (2.66), we get
dN(s;u)≤(1+2MeωT)e, ∀N ≥N1, s∈[0,T], u∈ Uad, which leads to
(2.71) sup
u∈Uad Z T
0 dN(s;u)ds≤ (1+2MeωT)Te, ∀N≥ N1. We consider now the term supu∈UadRT
0 deN(s;u)dswith deN(s;u) = sup
t∈[s,T]
k T(t−s)−eLN(t−s)ΠN
C(u(s))kH as defined in (2.20) for almost everysin[0,T]and everyuinUadhere.
Since the setU ⊂Vis compact (cf. Assumption(A5)) andC :V → His continuous, thenC(U)is a compact set ofH. Following a compactness argument similar to that used to derive (2.69), we can ensure the existence of an integerN2such that
(2.72) sup
t∈[0,T]
k T(t)−eLN(t)ΠN
C(w)kH≤ e, ∀N≥ N2, w∈U.
Now, for eachuinUad, sinceu(s)takes value inUfor almost everysin[0,T], we obtain from (2.72) that
(2.73) sup
t∈[s,T]
k T(t−s)−eLN(t−s)ΠN
C(u(s))kH≤e, for allN ≥N2and a.e. s∈[0,T]. It follows then
(2.74) Z T
0
deN(s;u)ds=
Z T
0 sup
t∈[s,T]
k T(t−s)−eLN(t−s)ΠN
C(u(s))kHds≤Te, ∀N≥ N2, u∈ Uad. Namely,
(2.75) sup
u∈Uad Z T
0
deN(s;u)ds≤ Te, ∀N ≥N2. The desired uniform convergence ofRT
0 dN(s;u)dsandRT
0 deN(s;u)dsfollows from (2.71) and (2.75).
The proof is now complete.
Remark 2.2. From the proof given above, it is clear that Assumption (A5) is made to ensure the uniform convergence ofRT
0 deN(s;u)ds, while Assumptions(A6)and(A7)are made to ensure the uniform convergence ofRT
0 dN(s;u)ds. The last two assumptions are thus not needed if the nonlinear term F is identically zero.
Note that one can readily check that the convergence result stated in Theorem2.1also holds when (2.1) is initialized at any other time instancet in [0,T). This will be needed in the next subsection to derive approximation results for value functions associated with optimal control problems for the IVP (2.1).
More precisely, for each(t,x)in[0,T)× H, we consider the following evolution problem (2.76)
dy
ds =Ly+F(y) +C(u(s)), s ∈(t,T], u∈ Uad[t,T], y(t) =x ∈ H,
with
(2.77) Uad[t,T]:= {u|[t,T] : u ∈ Uad}, and the corresponding Galerkin approximation:
(2.78)
dyN
ds = LNyN+ΠNF(yN) +ΠNC(u(s)), s∈ (t,T], u∈ Uad[t,T], yN(t) =xN, xN :=ΠNx ∈ HN.
Hereafter, we denote byyt,x(·;u)the solution to (2.76) emanating fromxat timet, and byyt,xNN(·;u) the solution to (2.78) emanating fromxN := ΠNxat timet. We have then the following corollary of Theorem2.1:
Corollary 2.1. Assume that the conditions of Theorem2.1hold. Then, for any mild solution yt,x(·;u)to(2.76) over[0,T], with x inHand u inUad[t,T]given in(2.77), the following convergence result is satisfied:
(2.79) lim
N→∞ sup
t∈[0,T]
sup
u∈Uad[t,T]
sup
s∈[t,T]
kyNt,xN(s;u)−yt,x(s;u)kH =0, where yNt,xN denotes the solution to the Galerkin approximation(2.78).
Proof. Since both the linear operatorLand the nonlinearityFin (2.76) are time independent, then the supremum of supu∈Uad[t,T]sups∈[t,T]kyNt,xN(s;u)−yt,x(s;u)kHastvaries in[0,T], is achieved att= 0, i.e.,
(2.80)
sup
t∈[0,T]
sup
u∈U[t,T]
sup
s∈[t,T]
kyNt,xN(s;u)−yt,x(s;u)kH = sup
u∈U[0,T]
sup
s∈[0,T]
ky0,xN N(s;u)−y0,x(s;u)kH
=sup
u∈U
sup
s∈[0,T]
kyN(s;xN,u)−y(s;x,u)kH. The desired estimate (2.79) follows then from (2.64).
2.5. Galerkin approximations of optimal control and value functions: Convergence results. We assume in this section thatUis a compact and convex subset of the Hilbert spaceV. In particular this ensures thatUaddefined in (2.32) is a bounded, closed and convex set. For such an admissible set of controls, conditions of existence to optimal control problems associated with the IVPs (2.1) and (2.8) are recalled in AppendixA.
We introduce next the cost functional,J: H × Uad →R+, associated with the IVP (2.1):
(2.81) J(x,u):=
Z T
0
[G(y(s;x,u)) +E(u(s))]ds, x ∈ H,